Uncertainty quantification (UQ) over graphs arises in a number of safety-critical applications in network science. The Gaussian process (GP), as a classical Bayesian framework for UQ, has been developed to handle graph-structured data by devising topology-aware kernel functions. However, such GP-based approaches are limited not only by the prohibitive computational complexity, but also the strict modeling assumptions that might yield poor coverage, especially with labels arriving on the fly. To effect scalability, we devise a novel graph-aware parametric GP model by leveraging the random feature (RF)-based kernel approximation, which is amenable to efficient recursive Bayesian model updates. To further allow for adaptivity, an ensemble of graph-aware RF-based scalable GPs have been leveraged, with per-GP weight adapted to data arriving incrementally. To ensure valid coverage with robustness to model mis-specification, we wed the GP-based set predictors with the online conformal prediction framework, which post-processes the prediction sets using adaptive thresholds. Experimental results the proposed method yields improved coverage and efficient prediction sets over existing baselines by adaptively ensembling the GP models and setting the key threshold parameters in CP.

Mixtures of linear dynamical systems (MoLDS) provide a path to model time-series data that exhibit diverse temporal dynamics across trajectories. However, its application remains challenging in complex and noisy settings, limiting its effectiveness for neural data analysis. Tensor-based moment methods can provide global identifiability guarantees for MoLDS, but their performance degrades under noise and complexity. Commonly used expectation-maximization (EM) methods offer flexibility in fitting latent models but are highly sensitive to initialization and prone to poor local minima. Here, we propose a tensor-based method that provides identifiability guarantees for learning MoLDS, which is followed by EM updates to combine the strengths of both approaches. The novelty in our approach lies in the construction of moment tensors using the input-output data to recover globally consistent estimates of mixture weights and system parameters. These estimates can then be refined through a Kalman EM algorithm, with closed-form updates for all LDS parameters. We validate our framework on synthetic benchmarks and real-world datasets. On synthetic data, the proposed Tensor-EM method achieves more reliable recovery and improved robustness compared to either pure tensor or randomly initialized EM methods. We then analyze neural recordings from the primate somatosensory cortex while a non-human primate performs reaches in different directions. Our method successfully models and clusters different conditions as separate subsystems, consistent with supervised single-LDS fits for each condition. Finally, we apply this approach to another neural dataset where monkeys perform a sequential reaching task. These results demonstrate that MoLDS provides an effective framework for modeling complex neural data, and that Tensor-EM is a reliable approach to MoLDS learning for these applications.

Out-of-Distribution (OOD) detection is critical to AI reliability and safety, yet in many practical settings, only a limited amount of training data is available. Bayesian Neural Networks (BNNs) are a promising class of model on which to base OOD detection, because they explicitly represent epistemic (i.e. model) uncertainty. In the small training data regime, BNNs are especially valuable because they can incorporate prior model information. We introduce a new family of Bayesian posthoc OOD scores based on expected logit vectors, and compare 5 Bayesian and 4 deterministic posthoc OOD scores. Experiments on MNIST and CIFAR-10 In-Distributions, with 5000 training samples or less, show that the Bayesian methods outperform corresponding deterministic methods.

Guiding pretrained flow-based generative models for conditional generation or to produce samples with desired target properties enables solving diverse tasks without retraining on paired data. We present ESS-Flow, a gradient-free method that leverages the typically Gaussian prior of the source distribution in flow-based models to perform Bayesian inference directly in the source space using Elliptical Slice Sampling. ESS-Flow only requires forward passes through the generative model and observation process, no gradient or Jacobian computations, and is applicable even when gradients are unreliable or unavailable, such as with simulation-based observations or quantization in the generation or observation process. We demonstrate its effectiveness on designing materials with desired target properties and predicting protein structures from sparse inter-residue distance measurements. In generative modeling, we are given data samples and aim to construct a sampler that approximates the data distribution. Diffusion models (Ho et al., 2020; Song et al., 2021) and continuous normalizing flows (Lipman et al., 2023; Liu et al., 2023; Albergo et al., 2023) achieve this by transporting samples from a simple source distribution to the data distribution.

Low rank inference on matrices is widely conducted by optimizing a cost function augmented with a penalty proportional to the nuclear norm $\Vert \cdot \Vert_*$. However, despite the assortment of computational methods for such problems, there is a surprising lack of understanding of the underlying probability distributions being referred to. In this article, we study the distribution with density $f(X)\propto e^{-λ\Vert X\Vert_*}$, finding many of its fundamental attributes to be analytically tractable via differential geometry. We use these facts to design an improved MCMC algorithm for low rank Bayesian inference as well as to learn the penalty parameter $λ$, obviating the need for hyperparameter tuning when this is difficult or impossible. Finally, we deploy these to improve the accuracy and efficiency of low rank Bayesian matrix denoising and completion algorithms in numerical experiments.

The Internet of Things (IoT) has revolutionized various applications including agriculture, but it still faces challenges in data collection and understanding. This paper proposes a real-time framework with three additional semantic layers to help IoT devices and sensors comprehend data meaning and source. The framework consists of six layers: perception, semantic annotation, interoperability, transportation, semantic reasoning, and application, suitable for dynamic environments. Sensors collect data in the form of voltage, which is then processed by microprocessors or microcontrollers in the semantic annotation and preprocessing layer. Metadata is added to the raw data, including the purpose, ID number, and application. Two semantic algorithms are proposed in the semantic interoperability and ontologies layer: the interoperability semantic algorithm for standardizing file types and the synonym identification algorithm for identifying synonyms. In the transportation layer, raw data and metadata are sent to other IoT devices or cloud computing platforms using techniques like WiFi, Zigbee networks, Bluetooth, and mobile communication networks. A semantic reasoning layer is proposed to infer new knowledge from the existing data, using fuzzy logic, Dempster-Shafer theory, and Bayesian networks. A Graphical User Interface (GUI) is proposed in the application layer to help users communicate with and monitor IoT sensors, devices, and new knowledge inferred. This framework provides a robust solution for managing IoT data, ensuring semantic completeness, and enabling real-time knowledge inference. The integration of uncertainty reasoning methods and semantic interoperability techniques makes this framework a valuable tool for advancing IoT applications in general and in agriculture in particular.

Large language models (LLMs) are increasingly used for decision-making tasks under uncertainty; however, their risk profiles and how they are influenced by prompting and alignment methods remain underexplored. Existing studies have primarily examined personality prompting or multi-agent interactions, leaving open the question of how post-training influences the risk behavior of LLMs. In this work, we propose a new pipeline for eliciting, steering, and modulating LLMs' risk profiles, drawing on tools from behavioral economics and finance. Using utility-theoretic models, we compare pre-trained, instruction-tuned, and RLHF-aligned LLMs, and find that while instruction-tuned models exhibit behaviors consistent with some standard utility formulations, pre-trained and RLHF-aligned models deviate more from any utility models fitted. We further evaluate modulation strategies, including prompt engineering, in-context learning, and post-training, and show that post-training provides the most stable and effective modulation of risk preference. Our findings provide insights into the risk profiles of different classes and stages of LLMs and demonstrate how post-training modulates these profiles, laying the groundwork for future research on behavioral alignment and risk-aware LLM design.

Catastrophic interference, also known as catastrophic forgetting, is a fundamental challenge in machine learning, where a trained learning model progressively loses performance on previously learned tasks when adapting to new ones. In this paper, we aim to better understand and model the catastrophic interference problem from a latent representation learning point of view, and propose a novel theoretical framework that formulates catastrophic interference as an identification problem. Our analysis demonstrates that the forgetting phenomenon can be quantified by the distance between partial-task aware (PTA) and all-task aware (ATA) setups. Building upon recent advances in identifiability theory, we prove that this distance can be minimized through identification of shared latent variables between these setups. When learning, we propose our method \ourmeos with two-stage training strategy: First, we employ maximum likelihood estimation to learn the latent representations from both PTA and ATA configurations. Subsequently, we optimize the KL divergence to identify and learn the shared latent variables. Through theoretical guarantee and empirical validations, we establish that identifying and learning these shared representations can effectively mitigate catastrophic interference in machine learning systems. Our approach provides both theoretical guarantees and practical performance improvements across both synthetic and benchmark datasets.

Learning about the directed acyclic graph (DAG) underlying a system's data-generating process from observational data under causal sufficiency is a fundamental causal discovery task (Pearl, 2009). Score-based algorithms address this task by assigning penalized likelihood scores to each DAG and seeking graphs whose scores are optimal. Identifiability theory asks when such score-optimal graphs identify the target graph (or its equivalence class) in the infinite-sample limit, with various results under different assumptions and scores (Chickering, 2002; Nandy et al., 2018). Exact algorithms, that are guaranteed to find a score-optimal graph, have exponential run-time and are feasible up to roughly 30 variables (Koivisto & Sood, 2004; Silander & Myllym aki, 2006). For larger graphs, local search must be employed, which evaluates neighbouring graphs to find graphs with better scores; canonical moves for this hill climbing are single edge insertions, deletions, or reversals (Heckerman et al., 1995). In the sample limit, greedy discrete search with a neighbourhood notion that respects score equivalence provably finds a graph with optimal score (Chickering, 2002). In finite samples, scores are inexact and local search may get stuck in local optima or, as we demonstrate, even find graphs with better scores than the true graph. Finite-sample performance is a practical challenge, despite the mature identifiability theory and asymptotic guarantees. Continuous optimization methods have emerged as a popular alternative.

Pass$@k$ is widely used to report performance for LLM reasoning, but it often yields unstable, misleading rankings, especially when the number of trials (samples) is limited and compute is constrained. We present a principled Bayesian evaluation framework that replaces Pass$@k$ and average accuracy over $N$ trials (avg$@N$) with posterior estimates of a model's underlying success probability and credible intervals, yielding stable rankings and a transparent decision rule for differences. Evaluation outcomes are modeled as categorical (not just 0/1) with a Dirichlet prior, giving closed-form expressions for the posterior mean and uncertainty of any weighted rubric and enabling the use of prior evidence when appropriate. Theoretically, under a uniform prior, the Bayesian posterior mean is order-equivalent to average accuracy (Pass$@1$), explaining its empirical robustness while adding principled uncertainty. Empirically, in simulations with known ground-truth success rates and on AIME'24/'25, HMMT'25, and BrUMO'25, the Bayesian/avg procedure achieves faster convergence and greater rank stability than Pass$@k$ and recent variants, enabling reliable comparisons at far smaller sample counts. The framework clarifies when observed gaps are statistically meaningful (non-overlapping credible intervals) versus noise, and it naturally extends to graded, rubric-based evaluations. Together, these results recommend replacing Pass$@k$ for LLM evaluation and ranking with a posterior-based, compute-efficient protocol that unifies binary and non-binary evaluation while making uncertainty explicit. Code is available at https://mohsenhariri.github.io/bayes-kit